Simplify the following expression and state the condition under which the simplification is valid. $r = \dfrac{t^3 - 14t^2 + 48t}{5t^2 - 40t + 60}$
First factor out the greatest common factors in the numerator and in the denominator. $ r = \dfrac {t(t^2 - 14t + 48)} {5(t^2 - 8t + 12)} $ $ r = \dfrac{t}{5} \cdot \dfrac{t^2 - 14t + 48}{t^2 - 8t + 12} $ Next factor the numerator and denominator. $ r = \dfrac{t}{5} \cdot \dfrac{(t - 6)(t - 8)}{(t - 6)(t - 2)}$ Assuming $t \neq 6$ , we can cancel the $t - 6$ $ r = \dfrac{t}{5} \cdot \dfrac{t - 8}{t - 2}$ Therefore: $ r = \dfrac{ t(t - 8)}{ 5(t - 2)}$, $t \neq 6$